How would an umbrella policy protect against a security deposit related claim?

Absolutely.If a landlord is listed as additional insured (rather than additional interest) they cannot make a claim against one another's liability policy for damages - it could completely jeopardize the coverage they think they're making it easier to get. 

So on numerous occasions, I have expressed my concerns about staying within my budget.Now on top of this, the property manager let the pipes freeze and they leaked and the boiler also cracked which I was told to file an insurance claim.

It protects both parties.Adding your property manager as an additional insured protects you and your property manager in case of a claim or lawsuit.

They claim it is a triplex (two regular units and an "efficiency unit".

We use the POA to allow for signing escrow refunds and claim checks.A couple other notes. 

They claim they own about 20 parks but a least 7 are owned by an investor and they just manage them.

You only want to file a claim if it's much greater than deductible or your rates shoot up pretty quickly. 

If the rents and upkeep are really as unreasonable as you claim there should be other cheaper and better solutions nearby. 

Investigate what (if any) additional fees that they charge for handling evictions, and find out whether they go to small-claims themselves, or hire an attorney.